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A053125 Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in decreasing order).

%I A053125 #23 Feb 16 2025 08:32:42
%S A053125 1,4,-2,16,-16,3,64,-96,40,-4,256,-512,336,-80,5,1024,-2560,2304,-896,
%T A053125 140,-6,4096,-12288,14080,-7680,2016,-224,7,16384,-57344,79872,-56320,
%U A053125 21120,-4032,336,-8,65536,-262144,430080,-372736,183040,-50688,7392,-480,9,262144,-1179648,2228224,-2293760,1397760
%N A053125 Triangle of coefficients of Chebyshev's U(n,2*x-1) polynomials (exponents of x in decreasing order).
%C A053125 A000302 (powers of 4), A002699, A002700 unsigned column sequences for m=0..2.
%C A053125 G.f. for row polynomials U(n,2*x-1) and row sums same as for A053124.
%C A053125 With offset 1 this is also the coefficient triangle of 2* U(2*n-1,x) expanded in decreasing powers of x. W. Lang, Mar 07 2007.
%D A053125 C. Lanczos, Applied Analysis. Prentice-Hall, Englewood Cliffs, NJ, 1956, p. 518.
%D A053125 Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
%H A053125 T. D. Noe, <a href="/A053125/b053125.txt">Rows n=0..50 of triangle, flattened</a>
%H A053125 W. Lang, <a href="/A053125/a053125.txt">First rows and related triangles </a>.
%H A053125 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChebyshevPolynomialoftheSecondKind.html">Chebyshev Polynomial of the Second Kind</a>
%H A053125 <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%F A053125 a(n, m) = A053124(n, n-m)= (4^(n-m))*A053123(n, m)= (4^(n-m))*((-1)^m)*binomial(2*n+1-m, m) if n >= m, else 0.
%F A053125 a(n, m) := -2*a(n-1, m-1)+4*a(n-1, m)-a(n-2, m-2), a(-2, m) := 0=: a(n, -2), a(-1, m) := 0=: a(n, -1), a(0, 0)=1, a(n, m)=0 if n<m;
%F A053125 G.f. for m-th column (signed triangle): ((-x)^m)*Po(m+1, 4*x)/(1-4*x)^(m+1), with Po(k, x) := sum('binomial(k, 2*j+1)*x^j', 'j'=0..floor(k/2)).
%e A053125 {1}; {4,-2}; {16,-16,3}; {64,-96,40,-4}; {256,-512,336,-80,5};... E.g. fourth row (n=3) corresponds to polynomial U^{*}(3,m)=U(3,2*x-1)= 64*x^3-96*x^2+40*x-4.
%t A053125 Reverse /@ CoefficientList[Table[ChebyshevU[n, 2 x - 1], {n, 0, 10}], x] // Flatten (* _Eric W. Weisstein_, Apr 04 2018 *)
%t A053125 Reverse /@ CoefficientList[ChebyshevU[Range[0, 10], 2 x - 1], x] // Flatten (* _Eric W. Weisstein_, Apr 04 2018 *)
%Y A053125 Cf. A053124, A053123.
%K A053125 easy,sign,tabl
%O A053125 0,2
%A A053125 _Wolfdieter Lang_